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Problem set 5
This homework is due on May 5th (Friday) 4pm. Please put your assignment into Yu’s mailbox.
1) (Overlapping Generations model) Consider an overlapping generations economy in which there
is one good in each period and each generation, except the initial old, lives for two periods. The
representative consumer in generation t, t = 1, 2, 3, ... has the following preferences
log ctt + β log ctt+1 ,
and the endowment (ett , ett+1 )=(1,1). The representative consumer in generation 0 (the initial old)
lives only in period 1, prefers more consumption to less, and has the endowment e01 =1. There is no
outside money (m = 0).
a) Define an Arrow-Debreu (AD) equilibrium for this economy. Calculate the unique AD equilibrium.
b) Suppose that β =2. Is the equilibrium allocation in part (a) Pareto efficient? If it is Pareto efficient,
prove that it is. If not, prove that it is not.
c) Suppose instead that β =0.8. Is the equilibrium allocation in part (a) Pareto efficient? If is Pareto
efficient, prove that it is. If not, prove that it is not.
d) Suppose now that there are equal numbers of two types of consumers in each generation. The
representative consumer of type i, i = 1, 2, in generation t, t=1,2,... has the following preferences
i
it
log cit
t + β log ct+1 ,
it
1
and the endowment (eit
t , et+1 )=(1,1). The two types differ only in their discount rates: β = 2 and
β 2 = 0.8. The representative consumer of each type in generation 0 lives only in period 1, prefers
more consumption to less, and has the endowment ei0
1 =1. There is no outside money (m = 0).
Define an Arrow-Debreu equilibrium for this economy.
e) In the equilibrium of part (d), is it the case that c1t
t = 1? Explain why or why not.
2) (Income Uncertainty) Consider a small open economy inhabited by a continuum of infinitely lived
agents with common utility given by:
E0
∞
X
β t u(ct ),
t=0
1
u(c) = − e−γc , γ > 0,
γ
where β < 1 is the discount factor and c is consumption. Assume each agent faces the following
intertemporal budget constraint
At+1 = (1 + r)(At + yt − ct ),
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where A is wealth (assets), y is a stochastic endowment, r is the world interest rate. Assume
yt = y + t , where t is i.i.d. across time and agents and distributed normally with mean 0 and
variance σ 2 . Assume that the world interest rate is such that β(1 + r) = 1. Answer the following
sub-questions.
a) Show that the consumption function,
ct =
r
1
(At + yt + y) − π,
1+r
r
where π is a constant which depends only on γ, r, σ 2 , implies
ct − ct−1 = ∆ct =
r
(yt − y) + rπ.
1+r
b) Use your results in part (2a) to show that the consumption function in part (2a) satisfies the
agent’s intertemporal Euler equation, solve for the constant π and show that π > 0.
c) Show that average income and cross-sectional income variance are stationary but average consumption and cross-sectional consumption variance are not.
3) (Incomplete Market model) Consider an economy in which different individuals face idiosyncratic
shocks to their income. Assume quadratic preferences and β(1 + r) = 1. Suppose real (log) income
Y can be decomposed into a deterministic and a stochastic components and the stochastic component
can be further decomposed into a permanent P and a transitory component v . The income process
for each individual i is
Yi,t = f (xi ) + Pi,t + vi,t ,
where t indexes time and f (xi ) is a set of deterministic components of income (for example,
demographic, education, among others). We assume that the permanent component Pi,t follows
martingale process of the form
Pi,t = Pi,t−1 + φi,t ,
where φi,t is serially uncorrelated and i.i.d. across time and agents, and the transitory component
vi,t follows and M A(q) process, where the order q is to be established empirically:
vi,t =
q
X
θj i,t−j
j=0
with θ0 ≡ 1 and v is serially uncorrelated. It follows that (unexplained) income growth is
∆yi,t = φi,t + ∆vi,t ,
where yi,t = Yi,t − f (xi ). Finally assume that φi,t and i,t are mutually uncorrelated processes.
a) Show that ∆ci,t = α1 φi,t + α2 i,t , where 0 ≤ α1 , α2 ≤ 1.
b) Calculate cov(∆yt , ∆yt+s ), where s ≥ 0.
c) Calculate cov(∆ct , ∆ct+s ), where s ≥ 0.
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d) Calculate ∆var(∆ct ).
e) Calculate cov(∆ct , ∆yt+s ), where s ≥ 0.
Note that when you calculate cross-sectional variances and covariances the index i is consequently
omitted.